The Gaussian is infinitely flat at infinity. Equivalently, the
Maclaurin expansion (Taylor expansion about
) of

(D.3)

is zero for all orders. Thus, even though
is
differentiable of all orders at
, its series expansion fails to
approach the function. This happens because
has an
essential singularity at
(also called a
``non-removable singularity''). One can think of an essential
singularity as an infinite number of poles piled up at the same
point (
for
). Equivalently,
above has an
infinite number of zeros at
, leading to the problem with
Maclaurin series expansion. To prove this, one can show

(D.4)

for all
. This follows from the fact that exponential
growth or decay is faster than polynomial growth or decay. An
exponential can in fact be viewed as an infinite-order polynomial,
since

Another interesting mathematical property of essential singularities is
that near an essential singular point
the
inequality

(D.6)

is satisfied at some point
in every neighborhood of
, however small. In other words,
comes arbitrarily close
to every possible value in any neighborhood about an
essential singular point. This was first proved by Weierstrass
[42, p. 270].